⁴ In a pseudo-UP algebra X the following holds, for each x ∈ X , 1)

x · 0 = 0 and x ٨ 0 = 0 ,

⁹ A fuzzy subset κ is called fuzzy PUP subalgebra of X , if it satisfies the following axioms: i)

κ(x · y) ≥ min{κ(x), κ(y)} , for each x, y ∈ X,

⁹ A mapping κ: X → [0, 1] is a fuzzy PUP ideal of X , if i)

κ(0) ≥ κ(x),

⁸ Let κ and η be fuzzy subset of a set X and Y , respectively, the Cartesian product of κ and η is define by (κ × η)(x, y) = min{κ(x), η(y)} , for (x, y) ∈ X × Y .

⁸ If κ is a fuzzy subset of X , the strongest fuzzy relation on X , that is a fuzzy relation on κ is ξ defined by ξ(x, y) = min{κ(x), κ(y)} , for each x, y ∈ X .

⁹ Let f: X → Y be a homomorphism of a PUP algebra. If κ be a fuzzy PUP ideal of Y . Then f ⁻¹(κ) is a fuzzy PUP ideal of X

⁹ Let (X, ·, ٨, 0) and (Y, ·, ٨, 0) be a PUP algebra. f: X → Y is surjective and κ is a fuzzy PUP ideals of X . Then f(κ) is a fuzzy pseudo-UP ideals of Y , provided that the sup property holds.

Let X and Y be pseudo-UP algebras, we define on X ×Y by, For every (x, y), (u, v) ∈ X × Y , (x, y) · (u, v) = (x · u, y · v) , (x, y) ٨ (u, v) = (x٨, u, y ٨ v) , then clearly, (X × Y, ·, ٨, (0, 0)) is a pseudo-UP algebra.

Let κ be a fuzzy PUP ideal of X and α ∈ [0, 1]. Then we need to show that κ ^α is a fuzzy pseudo-UP ideal of X. Since κ is a fuzzy PUP ideal, we have κ(0) ≥ κ(x) ⇒ κ ^α (0) = min{α, κ(0)} ≥ min{α, κ(x)} = κ ^α (x), for all α ∈ [0, 1]. Also, κ is a fuzzy PUP ideal of gives the

κ(x · z) ≥ min{κ(x · (y ٨ z)), κ(y)} , for all x, y, z ∈ X. κ α ( x ⋅ z ) = min { α , κ ( x ⋅ z ) } ≥ min { α , min { κ ( x ⋅ ( y ٨ z ) ) , κ ( y ) } } = min { min { α , κ ( x ⋅ ( y ٨ z ) ) , min { α , κ ( y ) } } = min { κ α ( x ⋅ ( y ٨ z ) ) , κ α ( y ) } .

And, we have κ(x ٨ z) ≥ min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X. κ α ( x ٨ z ) = min { α , κ ( x ٨ z ) } ≥ min { α , min { κ ( x ٨ ( y ⋅ z ) ) , κ ( y ) } } = min { min { α , κ ( x ٨ ( y ⋅ z ) ) , min { α , κ ( y ) } } = min { κ α ( x ٨ ( y ⋅ z ) ) , κ α ( y ) } .

This shows that, κ ^α is a fuzzy PUP ideal of X. Since this is true for all α ∈ [0, 1], κ ^α is a fuzzy PUP ideal of X. □

The converse of above theorem need not be necessarily true. This fact is shown by the following example.

Let X = {0, a, b, c} be a set with binary operations “ ·” and “٨” defined by the following cayley table: Table 1.

·	0	a	b	c
0	0	a	b	c
a	0	0	b	c
b	0	a	0	c
c	0	a	0	0

Table 2.

٨	0	a	b	c
0	0	a	b	c
a	0	0	b	c
b	0	a	0	c
c	0	a	b	0

See ⁴ (X, ⋅ , ∗ , 0 ) is PUP algebra.

Define the fuzzy set κ ( x ) = { 0.7 , if x = 0 0.8 , if x = a 0.5 , if x = b 0.4 , if x = c .

Since κ(0) = 0.7 < 0.8 = κ(a) . It follows that κ is not a fuzzy PUP ideal of a PUP algebra X since it does not meet the requirements of the Definition 2.4 (i). If α = 0.4 , then κ ^α (x) = 0.4 , for all x ∈ X . Accordingly, using routine calculations, we determine κ ^α is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X .

If κ ^c is a fuzzy PUP ideal of a PUP algebra X and a fuzzy subset κ ^ν (x) = min{1 −κ(x), 1 − ν} , for all x ∈ X and ν ∈ [0, 1] is a fuzzy PUP ideal of X .

Let κ ^c be a fuzzy pseudo-UP ideal of X. Define κ ^ν (x) = min{1 − κ(x), 1 − ν} . Since κ ^c is a fuzzy PUP ideal of X, we have κ ^c (0) ≥ κ ^c (x), for all x ∈ X.

Then κ ^ν (0) = min{1 − κ(0), 1 − ν} ≥ min{1 − κ(x), 1 − ν} = κ ^ν (x), for all x ∈ X and ν ∈ [0, 1]. Next, suppose that κ ^c is a fuzzy PUP ideal of X that is κ ^c (x · z) ≥ min{κ ^c (x · (y ٨ z)), κ ^c (y)} and κ ^c (x ٨ z) ≥ min{κ ^c (x ٨ (y · z)), κ ^c (y)} , for all x, y, z ∈ X. Now, κ ν ( x ⋅ z ) = min { 1 − κ ( x ⋅ z ) , 1 − ν } = min { κ c ( x ⋅ z ) , 1 − ν } ≥ min { min { κ c ( x ⋅ ( y ٨ z ) ) , κ c ( y ) } , 1 − ν } = min { min { 1 − κ ( x ⋅ ( y ٨ z ) ) , 1 − ν } , min { 1 − κ ( y ) , 1 − ν } } = min { κ ν ( x ⋅ ( y ٨ z ) ) , κ ν ( y ) } . ⇒ κ ν ( x ⋅ z ) ≥ min { κ ν ( x ⋅ ( y ٨ z ) ) , κ ν ( y ) } .

Moreover, κ ^ν (x ٨ z) ≥ min{κ ^ν (x ٨ (y · z)), κ ^ν (y)} , for all x, y, z ∈ X and ν ∈ [0, 1]. κ ν ( x ٨ z ) = min { 1 − κ ( x ٨ z ) , 1 − ν } = min { κ c ( x ٨ z ) , 1 − ν } ≥ min { min { κ c ( x ٨ ( y ⋅ z ) ) , κ c ( y ) } , 1 − ν } = min { min { 1 − κ ( x ٨ ( y ⋅ z ) ) , 1 − ν } , min { 1 − κ ( y ) , 1 − ν } } = min { κ ν ( x ٨ ( y ⋅ z ) ) , κ ν ( y ) } . ⇒ κ ν ( x ٨ z ) ≥ min { κ ν ( x ٨ ( y ⋅ z ) ) , κ ν ( y ) } .

This implies that κ ^ν is a fuzzy PUP ideal of a PUP algebra X, for ν ∈ [0, 1]. □

This fact is evidenced by the following example, showing that the converse of the above theorem need not be true.

From Example 3.3 , we define the fuzzy subset κ ( x ) = { 0.7 , if x = 0 0.6 , if x = a 0.5 , if x = b 0.5 , otherwise .

Since κ ^c (0) = 0.2 < 0.3 = κ ^c (a) . It follows that κ ^c is not a fuzzy pseudo-UP ideal of X due to its failure to comply with the Definition 2.4 (i). If ν = 0.8 , then κ ^ν (x) = 0.2 , for all x ∈ X . Thus, through standard calculations, we obtain κ ^ν is a fuzzy pseudo-UP ideal of a pseudo-UP algebra X .

Let κ ^α and ξ ^α be a fuzzy PUP ideals of X . Then κ ^α ∩ ξ ^α is a fuzzy PUP ideal of X .

Let κ ^α and ξ ^α be a fuzzy PUP ideals of the universe X. First we need to show that (κ ∩ ξ) ^α = κ ^α ∩ ξ ^α. Now, ( κ ∩ ξ ) α ( x ) = min { α , ( κ ∩ ξ ) ( x ) } = min { α , min { κ ( x ) , ξ ( x ) } } = min { min { α , κ ( x ) } , min { α , ξ ( x ) } } = min { κ α ( x ) , ξ α ( x ) } = ( κ α ∩ ξ α ) ( x ) . ⇒ ( κ ∩ ξ ) α = κ α ∩ ξ α .

Then, ( κ ∩ ξ ) α ( 0 ) = min { α , ( κ ∩ ξ ) ( 0 ) } = min { min { α , κ ( x ) } , min { α , ξ ( x ) } } ≥ min { min { α , κ ( x ) } , min { α , ξ ) ( x ) } } = ( κ α ∪ ξ α ) ( x ) .

Next, ( κ ∩ ξ ) α ( x ⋅ z ) = min { α , ( κ ∩ ξ ) ( x ⋅ z ) } = min { κ α ( x ⋅ z ) , ξ α ( x ⋅ z ) } ≥ min { min { κ α ( x ⋅ ( y ٨ z ) ) , κ α ( y ) } , min { ξ α ( x ⋅ ( y ٨ z ) ) , ξ α ( y ) } } = min { min { κ α ( x ⋅ ( y ٨ z ) ) , ξ α ( x ⋅ ( y ٨ z ) ) } , min { κ α ( y ) , ξ α ( y ) } } = min { ( κ α ∩ ξ α ) ( x ⋅ ( y ٨ z ) ) , ( κ α ∩ ξ α ) ( y ) } .

Moreover, ( κ ∩ ξ ) α ( x ٨ z ) = min { α , ( κ ∩ ξ ) ( x ٨ z ) } = min { κ α ( x ٨ z ) , ξ α ( x ٨ z ) } ≥ min { min { κ α ( x ٨ ( y ⋅ z ) ) , κ α ( y ) } , min { ξ α ( x ٨ ( y ⋅ z ) ) , ξ α ( y ) } } = min { min { κ α ( x ٨ ( y ⋅ z ) ) , ξ α ( x ٨ ( y ⋅ z ) ) } , min { κ α ( y ) , ξ α ( y ) } } = min { ( κ α ∩ ξ α ) ( x ٨ ( y ⋅ z ) ) , ( κ α ∩ ξ α ) ( y ) } .

Consequently, κ ^α ∩ ξ ^α is a fuzzy PUP ideal of X. □

Let {κ ^α /i ∈ Λ} be a family of fuzzy PUP ideals of X . Then ∩ _i∈Λκ ^α is also fuzzy iiz PUP ideal of X .

Let κ ^α and ξ ^α be fuzzy PUP ideals of X . Then κ ^α ∪ ξ ^α may not qualify as a fuzzy PUP ideal of X .

Let X = {0, 1, 2, 3} be a set with a binary operation “ ·” and “*” defined by the following cayley table: Table 3.

·	0	1	2	3
0	0	1	2	3
1	0	0	3	3
2	0	1	0	0
3	0	1	3	0

Table 4.

٨	0	1	2	3
0	0	1	2	3
1	0	0	3	3
2	0	1	0	0
3	0	2	3	0

See ⁴ (X, ⋅ , ∗ , 0 ) is PUP algebra. κ ( x ) = { 0.9 , if x = 0 0.6 , if x = 1 0.5 , if x = 2 0.4 , if x = 3 . ξ ( x ) = { 1 , if x = 0 0.4 , if x = 1 0.3 , if x = 2 0.1 , if x = 3 .

Clearly, κ and ξ are fuzzy PUP ideals of X . Thus by Theorem 3.1 κ ^α and ξ ^α are fuzzy PUP ideals of X , for α ∈ [0, 1] . If α = 0.7 , then κ ^α and ξ ^α are given by κ α ( x ) = { 0.7 , if x = 0 0.5 , if x = 1 0.5 , if x = 2 0.4 , if x = 3 . ξ α ( x ) = { 0.6 , if x = 0 0.4 , if x = 1 0.3 , if x = 2 0.1 , if x = 3 .

Now, (κ ^α ∪ ξ ^α )(1 · 3) = max{κ ^α (1 · 3), ξ ^α (1 · 3)} = max{κ ^α (3), ξ ^α (3)} = max{0.4, 0.1} = 0.1 . Hence , ( κ α ∪ ξ α ) ( 1 ⋅ 3 ) = 0.1 ............. ( ٨ ) . ( κ α ∪ ξ α ) ( 1 ⋅ 3 ) = max { κ α ( 1 ⋅ 3 ) , ξ α ( 1 ⋅ 3 ) } ≥ max { min { κ α ( 1 ⋅ ( 2 ٨ 3 ) ) , κ α ( 2 ) } , min { ξ α ( 1 ⋅ ( 2 ٨ 3 ) ) , ξ α ( 2 ) } } = max { min { κ α ( 1 ⋅ 0 ) , ξ α ( 1 ⋅ 0 ) } , min { κ α ( 2 ) , ξ α ( 2 ) } } = max { min { κ α ( 0 ) , ξ α ( 0 ) } , min { κ α ( 2 ) , ξ α ( 2 ) } } = max { min { 0.7 , 0.7 } , min { 0.6 , 0.4 } } = max { 0.7 , 0.4 } = 0.7 .

From (٨) we get 0.1 ≥ 0.7 which contradicts the Definition 2.4 .

Any PUP subalgebra of a PUP algebra X can be realized as a level PUP subalgebra of some fuzzy PUP subalgebra of X .

Let κ be a fuzzy PUP subalgebra of a PUP algebra X . Two upper level PUP subalgebra U(κ, t) and U(κ, s) (with t < s ) of a fuzzy PUP subalgebra are equal if and only if there is no x ∈ X such that t ≤ κ(x) < s .

Let κ be a fuzzy PUP subalgebra of X. Suppose that U(κ, t) = U(κ, s). Then we claim that there is no x ∈ X such that t ≤ κ(x) < s.

Assume that there exist x ∈ X such that t ≤ κ(x) < s. ⇒ x ∈ U ( κ , t ) , but x / ∈ U ( κ , s ) ⇒ U ( κ , s ) is a proper subset of U ( κ , s ) .

This contradicts to the assumption that U(κ, t) = U(κ, s).

Conversely, suppose that there is no x ∈ X such that t ≤ κ(x) < s. Then we have to prove that U ( κ , t ) = U ( κ , s ) . Since t < s , we get U ( κ , s ) ⊆ U ( κ , t ) (1)

Now, x ∈ U(κ, t) ⇒ κ(x) ≥ t ⇒ κ(x) > s (since κ(x) does not lie between t and s) ⇒ x ∈ U(κ, s) Hence , U ( κ , t ) ⊆ U ( κ , s ) (2)

From (1) and (2) we get U(κ, t) = U(κ, s). □

Let X be a finite PUP algebra and κ be a fuzzy PUP subalgebra of X . If Im (κ) = {t ₁, …, t _n } , then the family of PUP subalgebra { U (κ, t _i ) | 1 ≤ i ≤ n} , constitutes all the upper level PUP subalgebras of κ in X .

Let X be a PUP algebra and κ be a fuzzy PUP subalgebra of X . If Im(κ) is finite, say {t ₁, t ₂ …, t _n }, then for any t _i , t _j ∈ Im(κ), U (κ, t _i ) = U (κ, t _j ) ⇒ t _i = t _j.

Let κ and ξ be two fuzzy PUP subalgebra of a PUP algebra X with identical family of level PUP Subalgebras. If Im(κ) = {t ₁, t ₂ …, t _r} and Im(ξ) = {q ₁, q ₂ …, q _k} where t ₁ ≥ t ₂ ≥ ··· ≥ t _r and q ₁ ≥ q ₂≥ ··· ≥ q _k. Then 1)

k = r

Let κ and ξ be two fuzzy PUP subalgebras of a PUP algebra X with identical family of level PUP subalgebras. Then Im(κ) = Im(ξ) ⇒ κ = ξ .

Let Im(κ) = Im(ξ) = {q ₁, q ₂, …, q _r } where q ₁ ≥ q ₂ ≥ · · · ≥ q _r. By Theorem 3.15, for any x ∈ X there exists q _j such that κ(x) = q _i = ξ(x). Thus κ(x) = ξ(x) for all x ∈ X. This implies κ = ξ. □

Every PUP ideal of X can be realized as a level PUP ideal of some fuzzy PUP ideal of X .

Let J be a PUP ideal of a PUP algebra X and κ be a fuzzy subset in X defined by κ ( x ) = { t , if x ∈ J 0 , otherwise .

Clearly, U(κ, t) = J. Since 0 ∈ J, we have κ(0) = t. Thus κ(0) ≥ κ(x), for all x ∈ X. Now, consider the following cases:

Case i) If y ∈ J, x · (y ٨ z) ∈ J, then x · z ∈ J and x ٨ (y · z) ∈ J then x ٨ z ∈ J. Since J is a PUP ideal of X such that κ(x · z) = κ(y) = κ(x · (y ٨ z)) = t and κ(x ٨ z) = κ(y) = κ(x ٨ (y · z)) = t.

⇒ κ(x · z) = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) = min{κ(x ٨ (y · z)), κ(y)} .

Case ii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = t, κ(y) = 0 and κ(x ٨ (y · z)) = t. Then we have κ(x · z) ≥ 0 = min{0, t} = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{0, t} = min{κ(x ٨ (y · z)), κ(y)} .

Case iii) If x · (y ٨ z), x ٨ (y · z) ∈ J, y ∈ J, then κ(x · (y ٨ z)) = 0, κ(y) = t and κ(x ٨ (y · z)) = 0.

Then we get similar resualt as in case(ii).

Case iv) If y ∈ J, x · (y ٨ z), x ٨ (y · z) ∈ J, then κ(y) = 0 = κ(x · (y ٨ z)) = 0 and κ(x ٨ (y · z)) = 0.

⇒ κ(x · z) ≥ 0 = min{κ(x · (y ٨ z)), κ(y)} and κ(x ٨ z) ≥ 0 = min{κ(x ٨ (y · z)), κ(y)} , for all x, y, z ∈ X.

Thus κ is a fuzzy PUP ideal of a PUP algebra X. Hence, J is a level PUP ideal of X corresponding to a fuzzy PUP ideal of X. □

Let {J _i : i = 0, 1, 2, …, n} be any family of a PUP ideal of a PUP algebra X such that J ₀ ⊂ J ₁ ⊂ J ₂ ⊂ … ⊂ J _n = X , then there exist a fuzzy PUP ideal of X .

Let κ be a fuzzy PUP ideal of X , then the upper level PUP ideals U(κ, s) and U(κ, t) , with (s < t) of a fuzzy PUP ideals κ of X are equal if and only if there is no x ∈ X such that s ≤ κ(x) < t .

Consider κ is a fuzzy PUP ideal of X and U(κ, s) = U(κ, t), for some s < t. We need to show that there is no x ∈ X such that s ≤ κ(x) < t. Assume that there exist x ∈ x ∈ X such that s ≤ κ(x) < t. This implies that x ∈ U(κ, s) and x ∈ U(κ, t), which contradicts the assumption U(κ, s) = U(κ, t). Hence, there is no x ∈ X such that s ≤ κ(x) < t.

Conversely, assume that there is no x ∈ X such that s ≤ κ(x) < t. We need to prove that U(κ, s) = U(κ, t). Since s < t, we get U(κ, s) ⊆ U(κ, t) (٨).

Let x ∈ U(κ, s) then κ(x) ≥ s. By our assumption there is no x ∈ X such that κ(x) < t. Then κ(x) ≥ t ⇒ x ∈ U(κ, t) because κ(x) does not lie between s and t. Hence U(κ, s) ⊆ U(κ, t)................................ (٨ ٨).

From (٨) and (٨٨), it results in U(κ, s) = U(κ, t). □

Let κ be a fuzzy PUP ideal of X with finite image. If U(κ, s) = U(κ, t) , for some s, t ∈ Im(κ) , then s = t .

Let κ be a fuzzy PUP ideal of a PUP algebra X and let x ∈ X . Then κ(x) = s if and only if x ∈ U(κ, s) but x /∈ U(κ, t) , for all t > s .

Let (X, · _X , ٨ _X , 0 _X ) and (Y, · _Y , ٨ _Y , 0 _Y ) be PUP algebras. Let f: X → Y be homomorphism on PUP algebra. Then any fuzzy set κ in Y , we define a new fuzzy set κ ^f in X by κ ^f (x) = κ(f(x)).

Let f: X → Y be a homomorphism of PUP algebra. 1)

If a fuzzy subset κ is a fuzzy PUP subalgebra of Y , then a fuzzy set κ _f in X is a fuzzy PUP subalgebra of X .

1) Let f be a homomorphism of a PUP algebra and let κ be a fuzzy PUP subalgebra of Y. Let x, y ∈ X. Then κ f ( x ⋅ y ) = κ ( f ( x ⋅ y ) ) = κ ( f ( x ) ⋅ f ( y ) ) ≥ min { κ ( f ( x ) ) , κ ( f ( y ) ) } = min { κ f ( x ) , κ f ( y ) } . κ f ( x ٨ y ) = κ ( f ( x ٨ y ) ) = κ ( f ( x ) ٨ f ( y ) ) ≥ min { κ ( f ( x ) ) , κ ( f ( y ) ) } = min { κ f ( x ) , κ f ( y ) } .

Hence, κ ^f is a fuzzy PUP subalgebra of X.

2) Assume f is a homomorphism of a PUP algebras and κ is a fuzzy PUP ideal of X, then κ ^f (0) = κ(f(0)) = κ(0) ≥ κ(x) = κ(f(x)) = κ ^f (x), for all x ∈ X.

Additionally, let x, y, z ∈ X, then κ f ( x ⋅ z ) = κ ( f ( x ⋅ z ) ) = κ ( f ( x ) ⋅ f ( z ) ) ≥ min { κ ( f ( x ) ⋅ ( f ( y ) ٨ f ( z ) ) ) , κ ( f ( y ) ) } = min { κ ( f ( x ⋅ ( y ٨ z ) ) ) , κ ( f ( y ) ) } = min { κ f ( x ⋅ ( y ٨ z ) ) , κ ( f ( x ) ) } = min { κ f ( x ⋅ ( y ٨ z ) ) , κ f ( y ) } .

Moreover, κ f ( x ٨ z ) = κ ( f ( x ٨ z ) ) = κ ( f ( x ) ٨ f ( z ) ) ≥ min { κ ( f ( x ) ٨ ( f ( y ) ⋅ f ( z ) ) ) , κ ( f ( y ) ) } = min { κ ( f ( x ٨ ( y ⋅ z ) ) ) , κ ( f ( y ) ) } = min { κ f ( x ٨ ( y ⋅ z ) ) , κ ( f ( x ) ) } = min { κ f ( x ٨ ( y ⋅ z ) ) , κ f ( y ) } .

Therefore, κ ^f is a fuzzy PUP ideal of X. □

Let f: X → Y be an epimorphism of PUP algebra. Then κ ^f is a fuzzy PUP subalgebra of X if and only if κ is a fuzzy PUP subalgebra of Y .

Let f be a mapping defined on a PUP Algebra X . If κ is a fuzzy set in X , then the fuzzy set ξ in f(X) defined by ξ ( y ) = sup x ∈ f − 1 ( y ) κ ( x )

for all y ∈ f(X) is called image of κ under f . If ξ is a fuzzy set in f(X) , then the fuzzy set κ = ξ ◦f

in X i.e. the fuzzy set defined by κ(x) = ξ(f(x)) for all x ∈ X is called the pre-image of ξ under f .

Let f: X → Y be an epimorphism of PUP algebras. If ξ is a fuzzy PUP subalgebra of Y and let κ is the pre-image of ξ under f , then κ is a fuzzy PUP subalgebra of X .

Suppose ξ is a fuzzy PUP subalgebra of Y and κ is pre-image of ξ under f. For x, y ∈ X, we need to show that κ is a fuzzy PUP subalgebra of X. By Definition 3.25 κ = ξ ◦ f defined as κ(x) = ξ(f(x)). κ ( x ⋅ y ) = ξ ( f ( x ⋅ y ) ) = ξ ( f ( x ) ⋅ f ( y ) ) ≥ min { ξ ( f ( x ) ) , ξ ( f ( y ) ) } = min { κ ( x ) , κ ( y ) } . κ ( x ٨ y ) = ξ ( f ( x ٨ y ) ) = ξ ( f ( x ) ٨ f ( y ) ) ≥ min { ξ ( f ( x ) ) , ξ ( f ( y ) ) } = min { κ ( x ) , κ ( y ) } .

Hence, κ is a fuzzy PUP subalgebra of X. □

Let f: X → Y be an onto homomorphism of PUP algebra. If κ is a fuzzy PUP subalgebra of Y , then f ⁻¹(κ ^c ) = (f ⁻¹(κ)) ^c .

Let κ be a fuzzy PUP subalgebra of Y. Then, for x ∈ X. f − 1 ( κ c ) ( x ) = κ c ( f ( x ) ) = 1 − κ ( f ( x ) ) = 1 − f − 1 ( κ ) ( x ) = ( f − 1 ( κ ) ) c ( x ) . Thus f − 1 ( κ c ) = ( f − 1 ( κ ) ) c .

□

An ideal J of a PUP algebra X is said to be characteristic if f(J) = J , for all f ∈ Aut(X) , where Aut(X) is the set of all automorphisms of X . Similarly, a fuzzy PUP ideal κ of a PUP algebra X is said to be fuzzy characteristic if κ ^f (x) = κ(x) , for all x ∈ X and f ∈ Aut(X) .

A subset J of X is a PUP ideal if and only if the characteristic function of J is a fuzzy PUP ideal of X .

Suppose that κ is a fuzzy characteristic, and let s ∈ Im(κ), f ∈ Aut(X) and x ∈ U(κ, s). Then κ ^f ( x ) = κ ( x ) implies κ ( f ( x ) ) ≥ s , whence f ( x ) ∈ U ( κ , s ) .

Thus f ( U ( κ , s ) ) ⊆ U ( κ , s ) ( ٨ ) .

Let x ∈ U(κ, s) and y ∈ X such that f(y) = x. Then κ(y) = κ ^f (y) = mu(x) ≥ s. Consequnetly, y ∈ U(κ, s). So, x = f(y) ∈ U(κ, s).

Thus, U ( κ , s ) ⊆ f ( U ( κ , s ) ) ( ٨ ٨ ) .

From (٨) and (٨٨) we get f(U(κ, s)) = U(κ, s). Therefore, U(κ, s) is characteristic.

Conversely, suppose thta each level PUP ideal of κ is characteristic, and let x ∈ X, f ∈ Aut(X), κ(x) = s. Then by virtue of Theorem 3.21, x ∈ U(κ, s) and x ∈ U(κ, t), for all t > s. It follows from the assumption that f(x) ∈ f(U(κ, s)) = U(κ, s), so that κ ^f (x) = κ(f(x)) ≥ s. Let t = κ ^f (x) and assume that t > s. Then, f(x) ∈ U(κ, t) = f(U(κ, t)), which implies from the injectivety of f that x ∈ U(κ, t), a contradiction. Hence, κ ^f (x) = κ(f(x)) = s = κ(x) show that κ is a fuzzy characteristic. □

Let f: X → Y be an epimorphism of a PUP algebra. κ is a fuzzy PUP ideal of X with ker f ⊆ J , then f ⁻¹(f(κ)) = κ. □

Let y ∈ Y, then there exist x ∈ X such that f(x) = y. Then f ⁻¹(f(κ))(x) = f(κ)(f(x)) = κ ( y ) = sup x ∈ f − 1 ( y ) { κ ( x ) } . For any z ∈ X, z ∈ f ⁻¹(y) ⇒ f(z) = y ⇒ f(z) = f(x) implies f(z)·f(x) = 0 and f(z) ٨ f(x) = 0. Therefore, f(z · x) = 0 and f(z ٨ x) = 0, since f is homomorphism. ⇒ z ⋅ x , z ٨ x ∈ ker f ⇒ z ⋅ x , z ٨ x ∈ J , since ker f ⊆ J . ⇒ κ ( z ⋅ x ) = κ ( 0 ) and κ ( z ٨ x ) = κ ( 0 ) . ⇒ κ ( z ) ≥ min { κ ( z ⋅ x ) , κ ( x ) } = min { κ ( 0 ) , κ ( x ) } = mu ( x ) . ⇒ κ ( z ) ≥ min { κ ( z ٨ x ) , κ ( x ) } = min { κ ( 0 ) , κ ( x ) } = mu ( x ) . ⇒ κ ( z ) ≥ κ ( x ) .

Again we can prove that κ(x) ≥ κ(z) ⇒ κ(z) = κ(x). Thus, f − 1 ( f ( κ ) ) ( x ) = sup z ∈ f − 1 ( y )

{κ(z)} = κ(x), for some x ∈ X. Hence, f ⁻¹(f(κ)) = κ. □

Let f: X → Y be an epimorphism of PUP algebras and α ∈ [0, 1] . If κ ^α is a fuzzy PUP ideal of X , then f(κ ^α ) is a fuzzy PUP ideal of Y .

Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and α ∈ [0, 1]. If κ ^α is a fuzzy PUP ideal of Y , then the pre-image f ⁻¹(κ ^α ) is a fuzzy PUP ideal of X .

Let f: X → Y be an onto homomorphism of PUP algebras and κ ^α is a fuzzy subset in Y . If f ⁻¹(κ ^α ) is a fuzzy PUP ideal of X , then κ ^α is a fuzzy PUP ideal of Y , for α ∈ [0, 1].

Let f: X → Y be an epimorphism of PUP algebras, for all ν ∈ [0, 1] , and we define κ ^ν (x) = min{1 − κ(x), 1 − ν} is a fuzzy PUP ideal of X , then f(κ ^ν ) is a fuzzy PUP ideal of Y .

Since f: X → Y is an epimorphism of PUP algebras for any y ∈ Y, there exist x ∈ X such that f(x) = y. First we need to show that (f(κ)) ^ν = f(κ ^ν ). Now f ( κ ν ) ( y ) = sup { κ ν ( x ) : f ( x ) = y } = sup { min { 1 − κ ( x ) , 1 − ν } : f ( x ) = y } = min { sup { 1 − κ ( x ) , sup 1 − ν } } = min { 1 − f ( κ ) ( y ) , 1 − ν } = ( f ( κ ) ) ν , for all y ∈ Y .

Hence, (f(κ)) ^ν = f(κ ^ν ).

Then, (f(κ)) ^ν (0) = f(κ ^ν )(x) = f(κ ^ν )(f(0)) = f ⁻¹(f(κ ^ν ))(0) ≥ κ ^ν (0) ≥ κ ^ν (x), by Theorem 3.31 (٨). Implies κ ^ν (x) = f ⁻¹(f(κ ^ν ))(x) = f(κ ^ν )(f(x)) = f(κ ^ν )(y). Thus (f(κ)) ^ν (0) ≥ (f(κ)) ^ν (y), y ∈Y.

Also, let x, y, z ∈ Y, since f: X → Y is an epimorphisms of PUP algebras, there exist a, b, c ∈ X, such that f(a) = x, f(b) = y, f(c) = z. Then ( f ( κ ) ) ν ( x ⋅ z ) = f ( κ ν ) ( x ⋅ z ) = f ( κ ν ) ( f ( a ) ⋅ f ( c ) ) = f ( κ ν ) ( f ( a ⋅ c ) ) ≥ κ ν ( a ⋅ c ) , by Theorem 3.31 ( ٨ ) ≥ min { κ ν ( a ⋅ ( b ٨ c ) ) , κ ν ( b ) } = min { f − 1 ( f ( κ ν ) ) , f − 1 ( f ( κ ν ) ) ( b ) } = min { f ( κ ν ) ( f ( a ⋅ ( b ٨ c ) ) ) , f ( κ ν ) ( f ( b ) ) } = min { f ( κ ν ) ( f ( a ) ⋅ ( f ( b ) ٨ f ( c ) ) ) , f ( κ ν ) f ( b ) } = min { f ( κ ν ) ( x ⋅ ( y ٨ z ) ) , f ( κ ν ) ( y ) } = min { ( f ( κ ) ) ν ( x ⋅ ( y ٨ z ) ) , ( f ( κ ) ) ν ( y ) } .

And also, ( f ( κ ) ) ν ( x ٨ z ) = f ( κ ν ) ( x ٨ z ) = f ( κ ν ) ( f ( a ) ٨ f ( c ) ) = f ( κ ν ) ( f ( a ٨ c ) ) ≥ κ ν ( a ٨ c ) , by Theorem 3.31 ( ٨ ) ≥ min { κ ν ( a ٨ ( b ⋅ c ) ) , κ ν ( b ) } = min { f − 1 ( f ( κ ν ) ) , f − 1 ( f ( κ ν ) ) ( b ) } = min { f ( κ ν ) ( f ( a ٨ ( b ⋅ c ) ) ) , f ( κ ν ) ( f ( b ) ) } = min { f ( κ ν ) ( f ( a ) ٨ ( f ( b ) ⋅ f ( c ) ) ) , f ( κ ν ) f ( b ) } = min { f ( κ ν ) ( x ٨ ( y ⋅ z ) ) , f ( κ ν ) ( y ) } = min { ( f ( κ ) ) ν ( x ٨ ( y ⋅ z ) ) , ( f ( κ ) ) ν ( y ) } .

Hence, f(κ ^ν ) is a fuzzy PUP ideal of Y. □

Let f: X → Y be a homomorphism of fuzzy PUP algebra X into Y and ν ∈ [0, 1].

If κ ^ν is a fuzzy PUP ideal of Y , then the pre-image f ⁻¹(κ ^ν ) is a fuzzy PUP ideal of X .

Let κ and ξ be fuzzy sets on a PUP algebra X . Then 1.

κ × ξ is a fuzzy relation on X .

Let κ and ξ be any two fuzzy PUP subalgebras of X and Y , respectively. Then (κ × ξ)(0, 0) ≥ (κ × ξ)(x, y) , for all (x, y) ∈ X × Y.

Let (x, y) ∈ X × Y. Then by definition, it follows that (κ × ξ)(0, 0) = min{κ(0), ξ(0)} ≥ min{κ(x), ξ(y)} = (κ × ξ)(x, y). □

Let κ ₁ and κ ₁ be any two fuzzy PUP subalgebras of X ₁ and X ₂ , respectively. Then the cartesian product κ ₁ × κ ₂ of κ ₁ and κ ₂ defined as (κ ₁ × κ ₂)(x ₁, x ₂) = min{κ ₁(x ₁), κ ₂(x ₂)} is also a fuzzy PUP subalgebra of X ₁ × X ₂.

Let X = X ₁ × X ₂ and κ = κ ₁ × κ ₂. For x = (x ₁, x ₂), y = (y ₁, y ₂) ∈ X, we have κ ( x ⋅ y ) = κ ( ( x 1 , x 2 ) ⋅ ( y 1 , y 2 ) ) = κ ( x 1 ⋅ y 1 , x 2 ⋅ y 2 ) = ( κ 1 × κ 2 ) ( x 1 ⋅ y 1 , x 2 ⋅ y 2 ) = min { κ 1 ( x 1 ⋅ y 1 ) , κ 2 ( x 2 ⋅ y 2 ) } ≥ min { min { κ 1 ( x 1 ) , κ 1 ( y 1 ) } , min { κ 2 ( x 2 ) , κ 2 ( y 2 ) } } = min { min { κ 1 ( x 1 ) , κ 2 ( x 2 ) } , min { κ 1 ( y 1 ) , κ 2 ( y 2 ) } } = min { ( κ 1 × κ 2 ) ( x 1 , x 2 ) , ( κ 1 × κ 2 ) ( y 1 , y 2 ) } = min { κ ( x , y ) } .

Also, κ ( x ٨ y ) = κ ( ( x 1 , x 2 ) ٨ ( y 1 , y 2 ) ) = κ ( x 1 ٨ y 1 , x 2 ٨ y 2 ) = ( κ 1 × κ 2 ) ( x 1 ٨ y 1 , x 2 ٨ y 2 ) = min { κ 1 ( x 1 ٨ y 1 ) , κ 2 ( x 2 ٨ y 2 ) } ≥ min { min { κ 1 ( x 1 ) , κ 1 ( y 1 ) } , min { κ 2 ( x 2 ) , κ 2 ( y 2 ) } } = min { min { κ 1 ( x 1 ) , κ 2 ( x 2 ) } , min { κ 1 ( y 1 ) , κ 2 ( y 2 ) } } = min { ( κ 1 × κ 2 ) ( x 1 , x 2 ) , ( κ 1 × κ 2 ) ( y 1 , y 2 ) } = min { κ ( x , y ) } .

This shows that, κ ₁ × κ ₂ is a fuzzy PUP subalgebra of X ₁ × X ₂. □

Let κ and ξ be fuzzy subsets of the PUP algebras X and Y , respectively. Suppose 0 _X and 0 _Y are the constant elements of X and Y , respectively. If κ × ξ is a fuzzy PUP subalgebra of X × Y then at least one of the following two statements holds. i.

κ(x) ≤ ξ(0 _Y ) , for all x ∈ X

Let κ be a fuzzy subset of a PUP algebra X . Let ξ be a strong fuzzy relation on X . If ξ is a fuzzy PUP subalgebra of X × X , then κ(0) ≥ κ(x) , for all x ∈ X.

Since ξ is a fuzzy PUP subalgebra of X × X and it follows that ξ(x, y) = min{κ(x), κ(y)} . By Lemma 3.37 ξ(0, 0) ≥ ξ(x, x). Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □

Let κ be a fuzzy subset of a PUP algebra X and let ξ be the strongest fuzzy relation on a PUP algebra X . Then κ is a fuzzy PUP subalgebra of X if and only if ξ is a fuzzy PUP subalgebra of X × X.

If κ and ξ are fuzzy PUP ideals of X and Y , respectively, then κ × ξ is a fuzzy PUP ideal of X × Y .

Let κ and ξ be two fuzzy subsets of X and Y , resepecively such that κ × ξ is a fuzzy PUP ideal of κ × ξ of X × Y . Then either κ is a fuzzy PUP ideal of X or ξ is a fuzzy PUP ideal of Y .

Let κ and ξ be fuzzy subsets of X and Y such that κ × ξ is a fuzzy PUP ideal of X × Y . Then i)

either κ(0) ≥ κ(x) or ξ(0) ≥ ξ(y) , for all (x, y) ∈ X × Y.

Let κ × ξ be a fuzzy PUP ideal of X × Y. Then i)

Suppose that κ(0) < κ(x), for some x ∈ X and ξ(0) < ξ(y), for some y ∈ Y. Then (κ × ξ)(x, y) = min{κ(x), ξ(y)} > min{κ(0), ξ(0)} = (κ × ξ)(0, 0). This is a contradiction to the fact that κ × ξ is a fuzzy PUP ideal of X × Y.

Therefore, either κ(0) ≥ κ(x) or ξ(0) ≥ ξ(y), for all (x, y) ∈ X × Y.

Let κ and Q be fuzzy PUP ideals of X and Y , respectively. Then κ × Q is a fuzzy PUP ideal of X × Y if and only if each non-empty upper t -level set U(κ × Q, t) is a PUP ideal of X × Y , for all t ∈ [0, 1] .

Let κ be a fuzzy set of X and let ξ be the strongest fuzzy relation on X . If ξ , we define ξ(x, y) = min{κ(x), κ(y)} is a fuzzy PUP ideal X × X , then κ(0) ≥ κ(x) , for all x, y ∈ X .

Suppose ξ is a fuzzy PUP ideal of X × X, it follows that ξ(0, 0) ≥ ξ(x, x), for all x ∈ X. Then we have κ(0) = min{κ(0), κ(0)} = ξ(0, 0) ≥ ξ(x, x) = min{κ(x), κ(x)} = κ(x). Hence, κ(0) ≥ κ(x), for all x ∈ X. □

Let κ be a fuzzy subset of X and let ξ be the strongest fuzzy relation on X , then κ is a fuzzy PUP ideal of X if and only if ξ is a fuzzy PUP ideal of X × X .